The invention relates generally to 2-dimensional (2D) gas concentration map construction. More specifically, the invention relates to a Gaussian Process (GP) that propagates data from individual projection path average measurement values to all pixels in a 2D map and models the global correlation among all pixels.
Tunable Diode Laser Absorption Spectroscopy (TDLAS) is a technique for measuring gas (e.g., O2, CO) concentrations and temperatures simultaneously for combustion systems such as a boiler. A TDLAS system measures an average value of a parameter over a predetermined projection path through a combustion region. One active research area in TDLAS is to reconstruct a 2D gas concentration map based on multiple projection path averages. A gas concentration map is useful in many applications such as combustion monitoring, diagnosis and optimization.
To construct a gas concentration map is a problem of 2D reconstruction from 1-dimensional (1D) projections, which is similar to the concept of Computed Tomography (CT) used in medical imaging. However, most widely used CT algorithms, such as filtered back projection, require many projections (multiple views and dense projections per view) to achieve a high resolution. In contrast, only a very small number of projection paths are typically set up on a boiler. For example, it is not uncommon that only five to ten projection paths are used. Additionally, a projection path may not be through an optimal location or at an optimal direction (view) because of physical restrictions or mounting difficulties. These challenges result in a large percentage of pixels in the map that are unobserved (not traversed) by a single projection path.
Since CT image reconstruction uses the Algebraic Reconstruction Technique (ART), ART would appear to be a natural fit because it can handle the issue of a small number of projections. However, for this extremely under-constrained problem (the number of unknown variables, gas concentrations in a 2D map, is far more than the number of available equations), ART may not be realizable.
Smoothness constraints among neighboring pixels are introduced as prior information to help address this under-constrained problem. Smoothness can be incorporated using smooth basis functions or bicubic spline interpolation as a post-processing step. However, smoothness constraints are local and cannot capture long range correlation among pixels. This requirement is difficult to meet due to the above challenges.
What is desired is a method and system that constructs a gas concentration map from 1D projection path averages.